1. Derivation of the Vector Dot Product and the Vector Cross Product

2. Derivation of the Vector Dot Product
u·v =∑i ui vi
= ∑i ui ei ∑i vj ej

3. (u1 e1 + u2 e2 + u3 e3) (v1 e1 + v2 e2 + v3 e3)
Kronecker Delta
ei·ej = δij = when i = j
0 when i ≠ j

4. = u1 e1 v1 e1 + u1 e1 v2 e2 + u1 e1 v3e3 + u2 e2 v1 e1 + u2 e2 v2 e2 + u2 e2 v3 e3 + u3 e3 v1 e1 + u3 e3 v2 e2 + u3 e3 v3 e3

5. = u1v1e1e1+ u2v2e2e2+u3v3e3e3
= u1v1+ u2v2+u3v3

6. Vector Cross Product Einstein Notation
u × υ = εijk e i uj υk  
= Σijkεijkeiujυk
= Σi Σj Σk εijkeiujυk

7. Levi-Civati Symbol ε = 0 unless i, j, k are distinct
+1 if i, j, k is an even permutation of (1, 2, 3)
-1 if i. j, k is an odd permutation of (1, 2, 3)
ε =

8. Derivation of the Cross Product
= (ε121 u2v1 + ε122 u2v2 + ε123 u2v3 + ε131 u3v1 + ε132 u3v2 + ε133 u3v3 ) e1+ (ε211 u1v1 + ε212 u1v2 + ε213 u1v3 + ε231 u3v1 + ε232 u3v2 + ε233 u3v3 )e2 + (ε311 u1v1 + ε312 u1v2 + ε313 u1v3 + ε321 u2v1 + ε322 u2v2 + ε323 u2v3 ) e3 

9. Levi-Civati Symbol
even 123, 231, odd 321, 213, 132

10. Derivation of the Cross Product
= (ε123 u2v3+ ε132u3v2) e1 + (ε213 u1v3 + ε231 u3v1) e2+ (ε312 u1v2 + ε321 u2v1) e3   
= (u2v3 – u3v2)e1 + (u1v3 – u3v1)e2 + (u1v2 – u2v1)e3